SOCRATES PROGRAMME A EUROPEAN SUMMER SCHOOL AN INTENSIVE COURSE ON ‘ICT Tools on PV - Systems Engineering: Teaching & Learning’ 3 – 18 July, 2006 Technological Educational Institute of Patra
Design, Simulation and Performance of Reflecting Parabolic Solar Collector GEORGE BARAKOS Department of Mechanical Engineering Technological Educational Institute of Patra GR26334 GREECE
[email protected] Introduction: In this lecture a study of mini parabolic collectors systems is presented. The mini Parabolic Collector Systems (P.C.S.) are designed as to be integrated on the roofs of a residential buildings or in a larger scale on the roofs of bigger buildings, (offices, store houses, green farm houses). The common approach to locate collector’s arrays on roofs is to slope them at approximately the optimum solar collector angle. Then a detailed parametric analysis of the constructed parabolic trough-concentrating collectors (P.T.C.) by using a simulation method developed is made. The simulation procedure reveals the individual contributions of both direct and diffuse components of the solar radiation to the flux reaching the cylindrical absorber of the P.T.C. In addition, one may study the effects of various optical and thermal parameters of the collector’s performance.
Collector’s Geometric Elements: In this section an analysis of the geometrical characteristics of the mini parabolic solar collectors is made. The data of the mini parabolic system (fig.1) used for our study are provided. Let us suppose that the length, l=1250mm, is big enough to avoid end effects, as the image from the end of the trough is formed beyond the end of the receiver [1]. The length or aperture 2 y s , and the height h of the parabola are given as data input according to the designer. From the equation of the parabola one may obtain the focus length f.
f=
y2 4⋅z
Let the cylindrical absorber tube have diameter d. Here some definitions about linear concentrating systems or parabolic trough concentrating systems may be given [2, 3, 4].
1
Fig.1: Characteristics of the parabola
•
Aperture is the opening through which the solar radiation enters. AA’=2ys
•
Concentrator or optical system is the part of the collector that directs radiation onto the receiver.
•
Area concentration ratio C or geometric concentration ratio C is the ratio of the area of aperture to the area of the receiver.
•
Flux concentration ratio is defined as the ratio of the average energy flux on the receiver to that on the aperture.
•
Acceptance angle 2θα is the angular range over which all or almost all rays are accepted without moving all or part of the collector. The half acceptance angle for the parabolic trough collector is given at the position y by the formulae:
sin θ y =
d
(1)
y 2 2f 1 + 2f
This means that a ray hits the absorber if the angle of incidence is θ <
θy .
θ = θ y Is the angle at which the rays reach the receiver tube tangentially. θ y : Increases with y.
y = ± y S the above formulae gives θ α , the largest angle for which all incident rays are accepted. This acceptance, half – angle of the parabolic trough, can be written in terms of rim angle ϕ R and concentration For
ratio C.
2
sin θ α =
sin ϕ R πC
(2)
Rim angle φR. This angle is related to aperture and focal length f by
y ϕ 2y tan R = s = s or 2f 2 4f
8(f y s ) tan ϕ R = 2 16(f y s ) − 1
(3)
The linear concentrating systems to be used in this study will be fixed, integrated on the roof, oriented to “Non Track” the sun, the concentration ratios must be intermediate for the linear imaging collectors like trough parabolic concentrating system. For these systems of relatively low concentration ratio, part of the diffuse radiation is reflected to the receiver, with the amount depending on the acceptance angle of the concentrator. In general, concentrators with receivers much smaller than the aperture (big concentration ratio) are effective only on beam radiation.
Construction, material details and geometry of the concentrating collector system: In R.E.S. Laboratory in order to study the behavior of mini parabolic collector systems and compare theoretical and experimental results, a construction of two panels of mini parabolic collectors has been made. For the construction of the reflecting parabola aluminum foils type MIRO 27-Hochglanz with a selective surface from ALANOD company were used. The foil thickness is Sf = 0,5 mm, the absorber tube is copper painted black, the absorber tube thickness is St = 1 mm. As absorber, copper tubes with various external diameters (e.g. 10mm, 15mm, 18mm, 22mm), were used.
Let a reflecting parabola with an aperture 2ys , height h, which implies a focal distance
f = ys2 / 4h
The aluminum foil takes the form of the trough parabola via two base parabolas and two head parabolas made by C.N.C. machine (see figures 2a ,2b,2c and 2d).
Fig.2a, b: Head and base parabola.
3
Fig. 2c, d: CNC machine where our base and head parabola constructed
Those parabolas (base and head) are constructed from common aluminum foil of 2mm thickness. The base parabola using cutting tool with radius R must have: Aperture
:
bb = 2ys + 2Sf / sinψ – 2R/sinψ
Height
:
h b = h + Sf - R
(5)
fb = (bb/2)2/ 4hb
(6)
Focal distance :
(4)
The head parabola must have: Aperture
:
bh = 2ys + 2Sf / sinψ
Height
:
hh = h + R
Focal distance
:
(7) (8)
fh = (bh/2)2/ 4hh
(9)
The sinus ψ is due to the fact that the tangent to the parabola on the upper extreme A of the aperture, see fig. 8 comes from:
dz tan ψ = dy
A
2y = 4f
= A
y s 2f
(10)
The equation of the parabola in a (O, y, z) system is: y2 = 4 f z
(11)
The aluminum reflecting parabola foil for every channel of the P.C.S. will have a mean length equal to:
4h + b 2 + 16h 2 1 b m2 m m 2 2 Lm = b m + 16h m + ln m 2 8h m bm
(11)
where,
4
b m = 2y s +
Sf sin ψ
(12)
1 hm = h + Sf 2 For the overall dimensions of the parabolic trough collector see figure 3a and 3b.
Fig.3a: The parabolic trough collector of single aperture 100mm.
Fig.3b: The parabolic trough collector of single aperture 160mm.
5
Determination of Solar Radiation Parameters concerning Parabolic Collectors: The determination of the solar radiation intense (direct, diffuse and global), can be done experimentally, using pyranometers, or theoretically, using prediction methods. Solar vectors: In a specific instant time, fig. 4 shows the position of the sun in the O(x,y,z) orthogonal system, where (x, y) plane is the horizontal plane.
Fig.4: Sun’s position on O(x,y,z) coordinate system The sun’s unit vector in this system determined by:
n s = cos α cos γ s i + cos α sin γ s j + sin αk Where
(13)
i , j, k are the unit vectors of the axis x, y, z.
As the real situation is a parabolic concentrating system on the building’s roof with inclination to horizontal βo and orientation from south eo, the sun’s unit vector with reference to an O(x1, y1, z1) system with an axis along the absorber tube central line x1, the y1 normal to x1 and z1, where z1 axis is normal to the roofs plane, can be determined by an orthogonal transformation, see fig.5 Fig’s 4 and 5 shows the roof’s inclination angle βο, the roof’s orientation angle eo, the solar altitude α = 90 – θz, the solar azimuth angle γs and how one with two rotations, first around the y axis and then around the z axis, goes from O (x, y, z) system to the O (x1, y1, z1) orthogonal system.
Fig.5: Sun’s and collector’s position on O(x1 ,y1 ,z1 ) coordinate system
The sun’s unit vector as it is given by (13) in the O (x, y, z) system with transformation matrix using the Eulerian angles [5,6]:
l1 = cosβcose ,
m1 = sine ,
n1 = − sinβcose
l 2 = − sinecosβ, m2 = cose ,
n 2 = sinesinβ
l 3 = sinβ,
n 3 = cosβ
m3 = 0,
(14)
provides the sun’s unit vector in the O(x1,y1,z1) system
n s = (n s ) x1 i1 + (n s ) y1 j1 + (n s ) z1 k 1
(15)
The components of the solar unit vector in the coordinate system Ο (x1 ,y1 ,z1) are:
(n s ) x1 = (n s ⋅ i1 ) = + cos α cos e cos β cos γ s + cos α sin γ s sin e − sin α sin β cos e (n s ) y1 = (n s ⋅ j1 ) = + cos α sin γ s cos e − cos α cos γ s sin e cos β + sin e sin α sin β
(16)
(n s ) z1 = (n s ⋅ k 1 ) = + cos α cos γ s sin β + sin α cos β j1 and k 1 are the unit vectors of the axis x1, y1, z1 respectively:
Where i1 ,
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i1 = l 1 i + m 1 j + n 1 k = cos β cos e i + sin e j − sin β cos ek j1 = l 2 i + m 2 j + n 2 k = − cos β sin e i + cos e j + sin e sin βk k 1 = l 3 i + m 3 j + n 3 k = sin β i + cos βk
(
(17)
)
l 1 ,m 1 ,n 1 , (l 2 ,m 2 ,n 2 ) , (l 3 ,m 3 ,n 3 ) (14) are the direction cosines of the three unit vectors i1 , j1 , k 1 of the axis x1 ,y1 ,z1 respectively.[6] The parameters
Energy Consideration aspect in parabolic collectors: The useful thermal energy rate delivered by the collector is [3]:
Q u = Ac[I a − U L (Tc − Ta )] ,or Q u = FR A C [I a − U L (Tfi − Ta )]
(18)
The optical efficiency ηop is defined as the fraction of the solar radiation incident on the aperture of the collector, which is absorbed at the surface of the absorber tube.
ηop =
Ia AC I T A ap
Then,
(19)
[
]
Q u = ηop I T A ap − AcU L (Tc − Ta ) , or Q u = FR ηop I T A ap − A C U L (Tfi − Ta )
As the instantaneous collector’s thermal efficiency
(20)
ηp is defined as the ratio of the useful energy delivered to the
energy incident on the aperture.
ηp =
Qu I T A ap
(21)
Then, the instant thermal efficiency of the collector:
ηp = ηop −
U L (TC − Ta ) A U L (Tfi − Ta ) , or ηp = FR ηop − C IT A ap IT
(22)
From equation (20) it is clear that the losses to the surroundings from the absorber increase as the fluid inlet temperature increases and so thermal efficiency decreases. The efficiency ηp decreases significantly with the inlet fluid temperature, but this decrease is slightly non-linear. The non-linearity is due to the fact that the value of the overall loss coefficient increases slightly as Tfi increases. An increase in the mass flow rate of the thermic fluid, has as a result an increase to the heat transfer coefficient h cf and due to this the collector efficiency factor F ′ , and the collector heat removal factor FR and the efficiency increases. An increase in the concentration ratio C, by decreasing the size of the absorber tube, results an increase of the thermal efficiency, as the value of the optical efficiency changes very slightly. The loss from the absorber tube which are inversely proportional to C decreases and, hence, the thermal efficiency increases.
8
According to equation (22) the difference between the values of the optical efficiency efficiency
ηop and the thermal
ηp is a measure of the losses due to reradiation and convection from the absorber to the
surroundings.
IT Ia Ac
The solar radiation intense incident on the aperture The solar radiation intense absorbed by the absorber tube The absorber tube receiver surface A c = πD o l
Aap
The absorber surface
Do l UL Tc Ta Tfi FR
The absorber tube external diameter The length of the trough parabola The overall heat losses coefficient The receiver temperature The ambient temperature The inlet fluid temperature The collector’s heat removal factor
A ap = 2y s ⋅ l
Energy losses of concentrating collectors: The losses occurring in the collector may be distinguished, in optical and thermal. Optical losses are those, which occur in the path of the incident solar radiation before it is absorbed at the surface of the absorber tube. The optical losses can be estimated by simulations methods. Thermal losses are due to convection and reradiation from the absorber tube and conduction through the ends and supports. Thermal losses from the receiver must be calculated, usually in terms of energy loss coefficient U L , which is based on the area of the receiver. In principle, temperature gradients on the receiver can be accounted for by a flow factor in energy balance calculations.
FR the collector heat removal factor, to allow the use of inlet fluid temperatures
The generalized thermal analysis of a concentrating collector is similar to that of a flat plate collector. It is necessary to derive appropriate expressions for the collector efficiency factor F ′ , the energy loss coefficient U L and the collector heat removal factor FR . With calculated.
FR and U L known, the collector useful gain can be easily
For an uncovered cylindrical absorbing tube that might be used as a receiver in a linear concentrating system we calculate the thermal loss coefficient U L with the assumption that there are no temperature gradients around to the receiver tube. In order to have more accurate results, three models for the calculation of the heat loss coefficient UL has been developed.
Mathematical modelling simulation: A simulation procedure has been developed in “FORTRAN 4.0” to study theoretically mini parabolic collectors. The program chooses a series of equal space segment, Ns, on the parabola’s aperture where the solar beam radiation impinges (fig. 6). This number was set equal to 64or 128or 256or 512or 1028. The higher Ns are, the more accurate the results are and more execution true is required.
9
Fig.6: Reflecting parabola and absorber on the plane (
y 1 , z 1 ).
Fig. 6 shows the reflecting parabola and the absorber section in a plane parallel to y1, z1 at the position x1. The algebraic equation of this parabola is:
y = 4f (z 1 + f ) 2 1
Or
y 12 z1 = −f 4f
(23)
where f is the focal distance. The algebraic equation of the cylindrical absorber section is:
y 12 + z 12 = R 2 Where
(24)
R is the absorber radius
The program takes under consideration the direct and the diffuse component of solar radiation. Finally, it summarizes the effects of both components. The scanning starts from point A (0,− y s , z d ) and finishes at point A ′(0, y s , z d ) .
AA ′ is the aperture, the
parallel line to the y1 axis at distance z1 = zd.
Direct component of solar radiation: The sunrays are considered to be uniformly and parallel impinging on the parabola’s aperture AA ′ = 2 y s (fig.20). The sun beam is considered as scanning the aperture. The scheme is given in fig.7 from point A until point A ′ . For every point of the aperture,(Ns+1) points, the program initially checks if the beam hits the absorber directly. Then the point and angle of incidence on the absorber is determined. At the same time it calculates the power impinging on the relative absorber arch and its intensity [W/m2] is calculated. If the beam does not hit the absorber directly, the program checks if it intersects the parabola and then checks if the reflected ray hits the absorber. Then, it determines the point and angle of incidence on the absorber and calculates the power hitting on the absorber relative arch and it’s intense [W/m2].
10
An analytic investigation about the beam directly impinging on the absorber and also, then the beam incident on the absorber after reflection on the parabola’s surface are followed.
Beam directly impinging the absorber: The equation of the line that determines the unit vector of the sun which passes from an ordinary scanning point
n s on the coordinate system Ο(x1, y1, z1),
∆ ≡ (0, y 1∆ , z d ) of the aperture:
x 1 − 0 y 1 − y 1∆ z 1 − z d = = n x1 n y1 n z1
(25)
Solving the system of the cylinder equation (24) and the solar line equation (25) we check if the solar line intersects the cylinders surface at point C( x 1C , y 1C , z 1C ) on the cylinder. At position x 1 = x 1C we have a circular section (fig.7).
ns
z1
θid
C( x 1C , y1C , z1C )
R λ
y1
Fig.7: Beam directly impinging the absorber Parameters
n s and θ id are not necessarily located in the plane y1, z1.
tan λ =
z 1C y 1C
(26)
The program registers these angles λ.
The unit vector of the cylindrical radius
R = cos λ j1 + sin λk 1
R at point C is then given by:
if y 1C > 0
or (27)
R = − cos λ j1 − sin λk 1 if y 1C < 0
The angle of incidence of the direct beam on the absorber at point C is given by:
11
cos θ id = (n s ⋅ R ) = (n s ) y 1 cos λ + (n s ) z 1 sin λ if y 1C > 0
(28)
or
cos θ id = (n s ⋅ R ) = (n s ) y1 ( − cos λ ) + (n s ) z 1 ( − sin λ ) if y 1C < 0
Then the energy rate of the direct beam radiation incident directly and absorbed on the relative arch at point C is calculated:
Pbd = I b
cos θ ⋅ s ⋅ l ⋅ cos θ id ⋅ a sin α
[W]
or
Pbd = I b ⋅
where:
s=
cos θ 2y s ⋅ ⋅ l ⋅ cos θ id ⋅ a cos θ z N s
[W]
(29)
2y s is the step on the aperture 2y s . Ns
l = the length of the parabola trough a = receiver’s absorption.
The ratio of the power Pbd to the relative surface of the cylinder R ⋅ dλ ⋅ 1 gives the intense [W/m2]. Direct beam incident on the absorber after reflection on the parabola’s surface:
If the representative line of the sun’s vector does not intersect the cylinder’s surface then we solve the system of its equation (25) and the parabola’s equation (23). If the solar line intersects the parabola at a point E( x 1E , y1E , z1E ) we consider the parabola of fig.8 in the position x 1E . The equation of the tangent tt ′ to the parabola at the point E( y1E , z1E ) on a plane y1 , z1 at the abscissa x 1E is:
z 1 − z 1E = m(y 1 − y 1E ) where
dz 1 d y 12 − 4f 2 y 1E = = m = tan ψ = dy 1 E dy 1 4f 2f
(30)
12
Fig.8: Schematic representation of direct beam incident after reflection.
Then, the equation of the tangent at point E is:
z 1 = my 1 + b = z 1 =
y 1E y 1 − z 1E − 2f 2f
(31)
The normal to the tangent at point E is:
2f 1 z 1 = − y 1 + b = − y 1 − z 1E − 2f y 1E m
(32)
The unit vector of the normal to the parabola at point E is:
n p = − sin ψ j1 + cos ψ k 1 where For the reflection law:
where
ψ = arctan
y 1E 2f
θi = θr .
(
)
So,
cos θ i = cos θ r = n s ⋅ n p = (n s )y1 (− sin ψ ) + (n s )z1 cos ψ
(n s )y
and
1
(n s )z
1
(33)
obtained from equation (16)
The unit vector of the reflected ray is [21]:
r = −n s + 2(n s ⋅ n p )n p = −n s + 2 cos θ r n p
(34)
13
The reflected unit vector components are:
(r )x (r )y (r )z
1
= −(n s )x1
1
= −(ns )y 1 − 2 cos θ r sin ψ
1
= −(n s )z1 + 2 cos θ r cos ψ
The equation of the reflected solar radiation’s unit vector line from point E is:
x 1 − x 1E y 1 − y 1E z 1 − z 1 E = = (r )x1 (r )y1 (r )z1
(35)
To calculate the point where this line intersects the cylinder, we must solve the system of equations (25) and (35). We take into account only one reflection, (this is true for the trough parabolic concentrator).
If the reflected line intersects the cylinder at a point C( x 1C , y1C , z 1C ) then we have, equations (26) and (27)
tan λ =
z 1C , y 1C
R = ± cos λ j1 ± sin λ k 1
The angle of incidence of the reflected beam radiation on the absorber at point C is given:
(
)
(
)
cos θ ir = r ⋅ R = (r )y1 cos λ + (r )z1 sin λ , Or
if y 1C
>0
cos θ ir = r ⋅ R = (r )y1 (− cos λ ) + (r )z 1 (− sin λ ) , if y 1C < 0 Then, we calculate the energy rate of the direct beam radiation incident and absorbed after reflection on the relative arch to the point C.
Pbr = I b
cos θ ⋅ s ⋅ l ⋅ r ⋅ cos θ ir ⋅ a sin α
Or
(36)
cos θ 2y s Pbr = I b ⋅ ⋅ ⋅ l ⋅ r ⋅ cos θ ir ⋅ a cos θ z N s r = parabola’s reflectance The ratio of the power Pbr to the relative correspondent surface of the cylinder R ⋅ dλ ⋅ l gives the intense [w/m2]. Finally, the program stores the coordinates of the points of the absorber where the beam radiation has been absorbed. The profile of these points is shown in fig. 9.
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Fig. 9: Ray trace diagram for the direct beam insolation, at solar time11:00, grid Ns=16
In order to predict the performance of the mini parabolic collector, the beam absorbed has to be determined, always with the heat generated profile at the tubes surface.
Diffuse component of solar insolation: The diffuse component of the solar radiation is assumed to be isotropic. Under this hypothesis the fraction of isotropic diffuse radiation accepted by a solar collector of concentration C is I d C [14]. Α focusing parabola with small C has a small acceptance angle and the acceptance for diffuse radiation is somewhat smaller, [14]. For the case of the irradiance on a collector aperture one may accept: a.
If the collector has a high concentration ratio ( C ≥ 10 ) then it accepts only a negligible amount of diffuse radiation and the effective irradiance on the aperture in this case is therefore:
I eff = I bo
cos θ = I bo R b cos θ z
θ is the angle of incidence of the sun on the aperture.
b.
If the collector has a low concentration ratio C ≤ 10 and do not receives radiation reflected from the ground, the effective irradiance is [21]:
I eff = I bo R b +
I do C
(37)
Because of the predominance of near-forward scattering in the atmosphere, the sky radiation tends to be centred on the sun, and therefore the isotropic model underestimates the actual acceptance for diffuse radiation.
15
Hence, for the solar radiation accepted by a collector of concentration C is:
I real ≥ I bo R b +
I do C
To determine the points where the diffuse radiation hit the absorber either directly or reflectively and the optical and energy efficiency we use the Monte Carlo method.[8]
Fig.10: Isotropic diffuse radiation – Monte Carlo Techniques
ϕ R is uniformly distributed between the values 0 and 2π. For the angle θ R , in the isotropic hypothesis of the diffuse insolation we can’t say that θ R is uniformly distributed between the values π 0 and but cos θ R is uniformly distributed between the values 0 and 1. 2 In the isotropic model the angle
ϕ R and θ R we use a random number generator, which provides R 1 and R 2 , then ϕ R = 2π R 1 and cos θ R = 1 − R 2 [8].
To choose values for
→
The random diffuse isotropic radiation unit vector
→
n d = ∆Κ has the components on the coordinate
system x 1 , y 1 , z 1 :
n dx1 = sin θ R sin ϕ R (38)
n dy 1 = sin θ R cos ϕ R n dz 1 = cos θ
And the diffuse solar radiation equation line from point
∆ = (0, y 1∆ , z d ) of the aperture is:
16
x 1 − 0 y 1 − y 1∆ z 1 − z d = = n dx1 n dy 1 n dz 1
(39)
The program checks initially if the random diffuse ray presented by the above equation hits the absorber directly. Then it determines the point and angle of incidence on the absorber. If the diffuse solar equation line intersects the parabola the program checks the reflected diffuse rays that hit the absorber. The diffuse radiation directly impinging the absorber and the diffuse radiation incident on the absorber after reflection on the parabolas surface is investigated in the following section.
Diffuse radiation directly impinging the absorber: Solving the system of the cylinder equation (23) and the diffuse irradiation equation line (39) we check if the solar line intersects the cylinders surface at a point C d ( x 1Cd , y 1Cd , z 1Cd ) on the cylinder. At position x 1 = x 1Cd we have the circular section of fig. 11
nd z1
θidd
C d ( x 1Cd , y1Cd , z1Cd )
R λ
y1
Fig.11: Isotropic diffuse impinging the absorber directly.
The quantities of the diffuse solar radiation unit vector
(n d ) and the incident angle θ idd are not necessarily
located in the plane z 1 , y 1 . we have
→
tan λ =
z 1Cd y 1Cd
→
R = cos λ j1 + sin λk 1 if y 1Cd > 0 →
→
R = − cos λ j1 − sin λk 1 →
→
→
if
y 1Cd < 0 →
cos θ idd = (n d ⋅ R ) = (n d ) y 1 cos λ + (n d ) z 1 sin λ
if
y 1Cd > 0
if
y 1Cd < 0
or →
→
→
→
cos θ idd = (n d ⋅ R ) = (n d ) y1 ( − cos λ )+ (n d ) z 1 ( − sin λ )
17
And finally the energy rate of the diffuse radiation incident directly and absorbed on the relative arch at point C d is calculated.
Pdd = I d R d ⋅ The ratio of the power
2y s ⋅ l ⋅ cos θ idd ⋅ a [W] Ns
(40)
Pdd to the relative correspondent surface of the cylinder R ⋅ dλ ⋅ l gives the intense
[W/m2 ].
Diffuse radiation incident on the absorber after reflection on the parabola’s surface: If the solar line of the isotropic diffuse radiation does not intersect the cylinder’s surface, then we solve the system of equation (39) and the equation of the parabola (23). If the solar line intersects the parabola at a point E d ( x 1Ed , y 1Ed , z 1Ed ) we consider the parabola of fig.12 in the position x 1Ed .
Fig.12: Schematic representation of the diffuse insolation.
The equation of the tangent abscissa
tt ′ to the parabola at the point E d ( x 1Ed , y E1d , z 1Ed ) on a plane y 1 , z 1 at the
x 1Ed is:
z 1 − z 1 E q = m ( y 1 − y 1 Ed )
18
2 dz 1 d y 1 − 4f 2 y 1Ed m = tan ψ = = = dy 4f 2f dy 1 E
Where
d
Then, the equation of the tangent at point
E d is:
y 1E y 1 − z 1E d − 2f 2f The normal to the tangent at point E d is: z 1 = my 1 + b =
1 2f z 1 = − y 1 + b = − y − z 1Ed − 2f m y 1Ed 1 The unit vector of the normal to the parabola at point
E d is:
n p = − sin ψ j1 + cos ψ k 1 Where For the reflection law: θ i
ψ = arctan
y 1E d 2f
= θr .
(
)
cos θ i = cos θr = n d ⋅ n p = (n d ) y (− sin ψ) + (n d ) z cos ψ
So,
1
1
The unit vector of the reflected ray in the case of isotropic diffuse radiation is [21]:
(
)
rd = −n d + 2 n d ⋅ n p n p = −n d + 2 cos θr n p In this case, the reflected unit vector components are:
(r )x d = −(ns )x (r )y d = −(ns )y (r )z d = −(ns )z 1
1
1
1
1
1
− 2 cos θ r sin ψ + 2 cos θ r cos ψ
The equation of the reflected solar radiation’s unit vector line from point
x 1 − x 1Ed y 1 − y 1Ed z 1 − z 1Ed = = ( r) x1d ( r ) y 1d ( r ) z 1d
E d now is: (41)
Now, to determine the point where this line intersects the cylinder, we must solve the system of equations (23) and (41). We take into account only one reflection; this is true for the trough parabolic concentrator. If the reflected line intersects the cylinder at point
tan λ =
z 1Cd y 1Cd
C( x 1C d , y 1Cd , z 1Cd ) then we have, as in subchapter 5.3.1:
,
R = ± cos λ i1 ± sin λ k 1
19
cos θ ird = (rd ⋅ R ) = ( r ) y1d cos λ + ( r ) z 1d sin λ ,
if
y 1C d > 0
if
y 1Cd < 0
or
cos θ ird = (rd ⋅ R ) = ( r ) y1d ( − cos λ) + ( r ) z 1d ( − sin λ) , Finally, the energy rate the point
Pdr of the diffuse radiation incident after reflection and absorbed on the relative arch at
C d is calculated using the equation:
Pdr = I d ⋅ R d ⋅ The ratio of the power
2y s ⋅ l ⋅ cos θ ird ⋅ r ⋅ a Ns
(42)
Pdr to the relative correspondent surface of the cylinder R ⋅ dλ ⋅ l gives the intense
[W/m2]. The program stores the coordinates of the absorber’s points where the diffuse radiation has hit and gives the profile of these points (fig. 13)
a
b
Fig.13a, b: Ray trace diagram for diffuse radiation. Impinging the absorber (a) directly, (b) after reflection.
Estimation of the performance: The developed numerical code gives three options to determine the concentrating collector system performance: a) b) c)
an instantaneous efficiency from the number of rays incident in the absorber and that incident on the aperture and then the optical efficiency and the collector’s thermal instantaneous efficiency and the long – term performance of the system using a fluid tank.
Instantaneous efficiency based on number of rays: We consider a number, N tot , of rays impinging on the aperture of solar concentrator. Then, we investigate, what happens in the two cases of direct and diffuse components of the solar radiation: a. For the direct component:
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In this case, the program gives the number of rays incident directly, of rays which hit the absorber after reflection,
N bd , on the absorber and the number
N br , on the reflecting concentrator.
We can define efficiency for the direct component:
ηb =
Nbd + Nbr N b,tot
(43)
b. For the diffuse component: In this case, we consider a number of positions, Monte Carlo method, a number of rays, directly and a number,
N d,tot , on the aperture and the program gives, using the
N d,tot . From these rays, a number, N dd , impinges the absorber
N dr , impinges the absorber after reflection.
We can define the efficiency for the diffuse component:
ηd = The numbers
Ndd + Ndr N d,tot
(44)
N d,tot and N b,tot are equal to each other.
Finally, we can consider the total efficiency
η=
I bo R b I R ηb + bo d ηd IT IT
(45)
I bo R b is the fraction of the total radiation impinging on the aperture which consider the direct IT I do R d is the fraction of the total radiation impinging on the aperture which considers the component and IT Where
diffuse component. As it is well known,
I T = I bo R b + I do R d
Instantaneous Optical Efficiency: The optical efficiency is defined as the ratio between the energy rate absorbed [W] and the energy rate incident on the aperture I T A ap [W]: [2]
ηop =
Q abs I T A ap
(46)
The program calculates the energy rate absorbed in four steps: 1.
The energy rate Pbd of the direct beam radiation incident directly and absorbed on the receiver (29).
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2. 3. 4.
The energy rate Pbr of the direct beam radiation incident and absorbed on the receiver after reflection (36). The energy rate Pdd of the diffuse radiation incident directly and absorbed on the receiver (40). The energy rate Pdr of the diffuse radiation incident after reflection and absorbed on the receiver (42).
The program distinguishes between optical efficiency for the direct beam radiation and optical efficiency for the diffuse radiation. Ns
Ns
i =1
i =1
∑ Pbd + ∑ Pbr Then,
ηopb = Ns
Ns
∑P
dd
ηopd =
(47)
I bo R b A ap + ∑ Pdbr
i =1
i =1
(48)
I do R d A ap
The total optical efficiency:
η op =
total energy rate absorbed (diffuse and direct ) total energy rate incident on the aperture (diffuse and direct )
That is, k
∑ (P
bd
η op =
+ Pbr + Pda + Pdr )
i =1
(49)
(I bo R b + I do R d )A
Or
ηop = ηopb
I bo R b I R + ηopd do d IT IT
(50)
Notice: As far concerning the trough parabolic with a tracking system with reflectivity r and absorptivity α of the absorber for the direct beam radiation the optical efficiency η opb is approximately ηopb ≈ r ⋅ a we can check this with the program results only at noon
Instantaneous thermal collector efficiency: The simulation code estimates the instantaneous thermal efficiency using two different paths. The first one is via equation (21).
ηp =
Qu A ap I T
Where, ground reflected radiation is neglected and useful heat gain rate is:
ɺ C p (Tfo − Tfi ) [W] Qu = m
(51)
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ɺ : m Cp :
Tfo , Tfi :
Fluid mass flow rate [kg/sec] Specific heat capacity of the fluid [J/ kgK] Fluid outlet and inlet temperatures respectively
The second path for the estimation of the instantaneous thermal efficiency can be described by the equations (22)
ηp = ηop −
U L (Tc − Ta ) IT
,
A U (T − Ta ) ηp = FR ηop − C L fi A ap IT
Long – term performance: In order to have the option to calculate the collectors’ long - term performance we constructed a fluid tank. ɺ in the absorber tube is every time selected to take 54 liter of fluid per hour and m2 of The fluid flow rate mass m aperture surface
Fig. 14: Tank used on our experiment
ɺ = m Where And
54 ⋅ A ap tot ⋅ 3600 ⋅ 1000
[m3/s]
(52)
A ap tot = 2y s ⋅ l ⋅ n ch
n ch is the number of parabolas channels.
The volume of the tank (fig. 14) is equal to the volume of the fluid that comes throw the collector in an hour with the above flow rate.
ɺ ⋅ 3600 [m3] V=m
(53)
To execute the software, we supposed Tsi = Tfi = Tc = Ta then with iteration after the calculation of FR and UL we take.
Tc = Ta +
A cIa − Qu Qu , Tfo = + Tf i ɺ Cp AcU L m
(54)
And
Tf =
Qu + Tfi ɺ Cp 2m
The program calculate the tank’s useful energy rate
Q us = FR ' A c [I a − U L (Tsi − Ta )]
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We assume FR’= 0.9 FR
For a time period of an hour and for unit absorber surface we have:
Q us ⋅ 3600 = (mc p ) S (Tsf − Tfi ) Ac
(55)
And then
Tsf =
3600 ⋅ Q us + Tsi A c (54Cp )s
(56)
References: 1.
Duffie, J. A. and W. A Beckman, “Solar Energy Thermal Processes”, John Wiley and sons, New York (1991).
2. 3.
Rabl, A, “Active solar collectors and their applications”.Oxford University Press, New York, (1985) Kreith, F and J. F. Kreider, “principles of solar Engineering”, Mc Graw – Hill, New York (1978
4. 5. 6.
Suhas P. Sukhatme, “Solar Energy” Tata Mc Graw – Hill, New Delhi (1996) J.R. Sir H. W. Massey and H. Kestelman, ‘Ancillary Mathematics’, Sir Isaac Pitman & Son, London (1964). Grant R. Fowles and G.L.Cassiday “Analytical Mechanics” Saunders Golden Sunburst Series, Orlando (1993
7.
D. E. Prapas, “Optics of Parabolic –Trough Solar Energy Collectors Possessing Small Concentrating Ratios”, Solar Energy, v39, No 6 (1987).
8.
S. N. Kaplanis “ The Monte Carlo Method” , Thessaloniki, (1978), (in Greek)
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